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A survey on toric degenerations of projective varieties

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arxiv 2301.02545 v1 pith:CB3GGRMR submitted 2023-01-06 math.AG

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keywords mathbbnewtonpolytopeadapteddegenerationssurveytoricvaluations
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abstract

In this survey I summarize the constructions of toric degenerations obtained from valuations and Gr\"obner theory and describe in which sense they are equivalent. I show how adapted bases can be used to generalize the classical Newton polytope to what is called a $\mathbb B$-Newton polytope. The $\mathbb B$-Newton polytope determines the Newton--Okounkov polytopes of all Khovanskii-finite valuations sharing the adapted standard monomial basis $\mathbb B$.

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  1. The SagbiHomotopy.jl package for solving polynomial systems

    math.AG 2025-06 conditional novelty 5.0 of 10

    SagbiHomotopy.jl implements SAGBI-basis homotopy continuation, reducing path counts for horizontally parameterized polynomial systems compared with polyhedral homotopies.

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